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Co-investment with Payoff Sharing Benefit Operators and Users in Network Design

Mingjia He, Andrea Censi, Emilio Frazzoli, Gioele Zardini

TL;DR

This work addresses suboptimal outcomes in interdependent subnetworks by proposing an interactive network design framework that combines non-cooperative and cooperative game theory. It models a two-region mobility network with multimodal options, introduces a Nash Equilibrium criterion for non-cooperative decisions, and enables cooperation through a co-investment mechanism and a payoff-sharing mechanism based on Nash Bargaining Solution ($NBS$). The Sioux Falls case study demonstrates that carefully allocated co-investment can approach system-optimal performance, yielding substantial reductions in emissions and travel costs while increasing rail revenue, with heterogeneous-region scenarios showing broader gains. The framework offers practical guidance for regional authorities on when and how to cooperate and how to share benefits in multi-region transportation design. $NE$, $NBS$, and related payoff concepts underpin the cooperative structure and allocation rules, linking theoretical rigor to applied planning insights.

Abstract

Network-based complex systems are inherently interconnected, with the design and performance of subnetworks being interdependent. However, the decisions of self-interested operators may lead to suboptimal outcomes for users. In this paper, we consider the question of what cooperative mechanisms can benefit both operators and users simultaneously. We address this question in a game theoretical setting, integrating both non-cooperative and cooperative game theory. During the non-cooperative stage, subnetwork decision-makers strategically design their local networks. In the cooperative stage, the co-investment mechanism and the payoff-sharing mechanism are developed to enlarge collective benefits and fairly distribute them. A case study of the Sioux Falls network is conducted to demonstrate the efficiency of the proposed framework. The impact of this interactive network design on environmental sustainability, social welfare and economic efficiency is evaluated, along with an examination of scenarios involving regions with heterogeneous characteristics.

Co-investment with Payoff Sharing Benefit Operators and Users in Network Design

TL;DR

This work addresses suboptimal outcomes in interdependent subnetworks by proposing an interactive network design framework that combines non-cooperative and cooperative game theory. It models a two-region mobility network with multimodal options, introduces a Nash Equilibrium criterion for non-cooperative decisions, and enables cooperation through a co-investment mechanism and a payoff-sharing mechanism based on Nash Bargaining Solution (). The Sioux Falls case study demonstrates that carefully allocated co-investment can approach system-optimal performance, yielding substantial reductions in emissions and travel costs while increasing rail revenue, with heterogeneous-region scenarios showing broader gains. The framework offers practical guidance for regional authorities on when and how to cooperate and how to share benefits in multi-region transportation design. , , and related payoff concepts underpin the cooperative structure and allocation rules, linking theoretical rigor to applied planning insights.

Abstract

Network-based complex systems are inherently interconnected, with the design and performance of subnetworks being interdependent. However, the decisions of self-interested operators may lead to suboptimal outcomes for users. In this paper, we consider the question of what cooperative mechanisms can benefit both operators and users simultaneously. We address this question in a game theoretical setting, integrating both non-cooperative and cooperative game theory. During the non-cooperative stage, subnetwork decision-makers strategically design their local networks. In the cooperative stage, the co-investment mechanism and the payoff-sharing mechanism are developed to enlarge collective benefits and fairly distribute them. A case study of the Sioux Falls network is conducted to demonstrate the efficiency of the proposed framework. The impact of this interactive network design on environmental sustainability, social welfare and economic efficiency is evaluated, along with an examination of scenarios involving regions with heterogeneous characteristics.
Paper Structure (19 sections, 1 theorem, 14 equations, 6 figures, 2 tables)

This paper contains 19 sections, 1 theorem, 14 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related research
  3. Statement of Contribution
  4. Method
  5. Modeling the Mobility Network and Travel Demand
  6. Intra-regional and Inter-regional Trips
  7. Elastic Demand
  8. Interactive Network Design Problem
  9. The Condition to Cooperate
  10. Interactive Network Design Framework
  11. Model structure and Variables
  12. Performance metrics
  13. Design Stage 1 - Non-Cooperative Network Design
  14. Design Stage 2 - Cooperative Network Design
  15. Optimization Problem for Co-investment Mechanism
  16. ...and 4 more sections

Key Result

Lemma 1

The payoff mechanism can produce an unique optimal solution for payoff allocations by the optimization problem, only if

Figures (6)

  • Figure 1: The interactive network design framework, featuring a non-cooperative, as well as a cooperative phase.
  • Figure 2: The interconnected multimodal mobility network for Region 1 and Region 2.
  • Figure 3: Sioux Falls network, subdivided between Region 1 and 2.
  • Figure 4: Equilibrium solutions of interactive network design.
  • Figure 5: Co-investment ratio and improved performance.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Definition 1: Equilibrium of the interactive network design process
  • Definition 2: Feasible agreement
  • Remark
  • Remark
  • Definition 3
  • Lemma 1: Existence and uniqueness of optimal payoff allocations
  • proof